In artificial cycles like for example refrigeration cycle, because of irreversibilities extra energy is require to be provided so that the refrigerant is available at the same state as it was at the beginning. So where does extra energy for water cycle come from? Does it have zero irreversibility?
As has already been pointed out, the Sun provides the excess energy for the water cycle to continue indefinitely (or at least for as long as we have the Sun). But let's investigate this a bit further.
Even though it seems as if water in a closed system would be able to evaporate and condense on its own through the vapor pressure of the water, there are a few concerns with that view.
Follow the energy
Firstly, it takes energy to break the inter-molecular bonds (in the case of water, these bonds are called hydrogen bonds) between the molecules of the liquid. This will cause the liquid to cool down, and with that cooling the vapor pressure will drop until the atmosphere above the liquid is saturated.
But, "Aha! The energy taken from evaporation will be given back when the vapor condenses again!" you might say. While it is true that the evaporation energy will be given back as heat upon condensation, for there to be a "cycle", this condensation must happen at some other location (e.g. high up in the atmosphere), which means that energy will have moved from the liquid to the location of condensation. Or in other words the liquid is cooling down and the point of condensation is heating up. At some point these two locations will reach the same temperature; when that happens, the vapor pressure will be the same at the two points, and there will be no flow of vapor in either direction. The system has reached thermal equilibrium, and will no longer evolve.
Note that this would happen even if you let the (warm) condensed liquid flow back to the original reservoir, e.g. through gravity. Because for the vapor to condense, you need to pull energy out of the vapor which can only happen if you have a "source of coldness", i.e. something cold which will absorb the energy of condensation.
Then look at the entropy
At this point we have pretty much exhausted the energy argument, which incidentally is called the First law of thermodynamics (the total energy inside a closed system does not change). But we must have some more argument for why the system would stop, or why there are no mechanisms that would allow us to keep it running. We therefore have to turn to the Second law of thermodynamics, which says that the entropy inside a closed system will always increase (or at least not decrease). This is always the scary one, since the concept of entropy is hard to have a natural intuition for. But let's see what we can pull out if it anyway.
I previously said that energy was flowing with the vapor from the liquid reservoir to the condensation point, which would require a "source of coldness" at the point of condensation. Another way of saying that is to state that for the vapor cycle to run, you would need to have two different temperatures, with the reservoir having a higher temperature than the condensation point.
However, now with the help of the Second law of thermodynamics, the unforgiving increase in entropy would dictate that the system will evolve towards a state where the temperatures are the same throughout the whole system, i.e. thermal equilibrium, thus completely ending the flow of energy/vapor.
The water cycle in the atmosphere
Now then, why does the water cycle continue on Earth if it could not be sustained in a closed system?
The simple answer is that the Earth and its atmosphere is not a closed system. We have energy flowing in from the Sun's radiation and energy flowing out through the Earth's and the atmosphere's black-body radiation. (I'm ignoring the heat of the Earth core, to not make the reasoning overly complicated.) An indirect consequence of the Second law is that when we have a flow of energy/heat, i.e. a temperature gradient, we can extract some of that energy in a useful form (called "work").
So if we examine the situation in our atmosphere, we find that the short wavelengths of the sunlight can penetrate through the atmosphere and heat up the land and waters on the surface of Earth. This will provide enough energy for water to evaporate without the reservoir getting cooler and cooler.
Then the water vapor rises to a higher layer of the atmosphere, which has not been heated up by the sunlight. However, the atmosphere still radiates black-body radiation which provides us with our "source of coldness" for the vapor to again condense. From here on, the story is familiar: the condensed water eventually rains back down on us and the cycle continues.
Thermodynamics requires a flow of heat to extract the work required to sustain the water cycle. This heat flow is provided to us by the Sun: we are tapping into the minuscule flow of heat that hits the Earth on its way from the Sun into outer space.
The cycle requires a source of energy---this is the Sun---and also a way to get rid of entropy---this is done by heat radiation into space.
The water cycle involves irreversibility at various stages, for example in turbulence in the atmosphere, rivers and sea; in plant chemistry; in heat flow across finite temperature differences, and so on. Therefore entropy is increasing. This entropy is eventually radiated away into space via infrared radiation from the planet. Overall this involves flow of both energy and entropy, as follows.
The incident sunlight is mainly in the visible part of the electromagnetic spectrum, while the radiation emitted by the Earth is mainly infrared. The crucial fact here is that in thermal radiation the amount of entropy per joule increases with the wavelength, so when a given amount of energy is first absorbed in visible light and then later emitted in infrared radiation, overall the energy balances out while more entropy is emitted than was absorbed. This process allows the Earth to get rid of entropy without losing energy overall and cooling down. This allows it to support processes such as the water cycle.
(I added this answer because I judged existing answers to be either too long or too short; but I think we all agree on the physics.)